Idempotent/tropical analysis, the Hamilton-Jacobi and Bellman equations

TL;DR

This paper explores the connections between tropical and idempotent analysis and classical equations like Hamilton-Jacobi and Bellman, highlighting dequantization procedures and their implications for optimization and numerical algorithms.

Contribution

It demonstrates how the Hamilton-Jacobi-Bellman equation arises from dequantization of the Schrödinger equation and discusses universal algorithms in idempotent mathematics.

Findings

Hamilton-Jacobi-Bellman equation is linear over tropical algebras.

Dequantization links Schrödinger and Hamilton-Jacobi equations.

Universal algorithms for idempotent numerical analysis are examined.

Abstract

Tropical and idempotent analysis with their relations to the Hamilton-Jacobi and matrix Bellman equations are discussed. Some dequantization procedures are important in tropical and idempotent mathematics. In particular, the Hamilton-Jacobi-Bellman equation is treated as a result of the Maslov dequantization applied to the Schr\"{o}dinger equation. This leads to a linearity of the Hamilton-Jacobi-Bellman equation over tropical algebras. The correspondence principle and the superposition principle of idempotent mathematics are formulated and examined. The matrix Bellman equation and its applications to optimization problems on graphs are discussed. Universal algorithms for numerical algorithms in idempotent mathematics are investigated. In particular, an idempotent version of interval analysis is briefly discussed.